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We calculate the complete one-loop electroweak corrections to the inclusive$H\rightarrow ZZ$ and $H\rightarrow Z\gamma$ decays in the dimension-$6$extension of the Standard Model Effective Field Theory (SMEFT). The correctionsto $H\rightarrow ZZ$ are computed for on-shell $Z$ bosons and are a precursorto the physical $H\rightarrow Z f {\overline{f}}$ calculation. We presentcompact numerical formulas for our results and demonstrate that the logarithmiccontributions that result from the renormalization group evolution of the SMEFTcoefficients are larger than the finite NLO contributions to the decay widths.As a by-product of our calculation, we obtain the first complete result for thefinite corrections to $G_\mu$ in the SMEFT.

Higgs Decays to $ZZ$ and $Z\gamma$ in the SMEFT: an NLO analysis